An elementary proof of a conjecture of Saikia on congruences for \(t\)-colored overpartitions (Q6189056)

The starting point for this work is the family of functions \(p_{-t}(n)\) which counts the number of \(t\)-colored overpartitions of \(n\). A very elegant statement is proved: For all \(n\geq 0\) and all odd \(t\), \begin{align*} p_{-t} (8n + 1) \equiv 0 \pmod 2, \\ p_{-t} (8n + 2) \equiv 0 \pmod 4, \\ p_{-t} (8n + 3) \equiv 0 \pmod 8, \\ p_{-t} (8n + 4) \equiv 0 \pmod 2, \\ p_{-t} (8n + 5) \equiv 0 \pmod 8, \\ p_{-t} (8n + 6) \equiv 0 \pmod 8, \\ p_{-t} (8n + 7) \equiv 0 \pmod{32}. \end{align*}